3.5.70 \(\int \frac {x (c+d x)^{5/2}}{(a+b x)^2} \, dx\) [470]

Optimal. Leaf size=178 \[ \frac {(2 b c-7 a d) (b c-a d) \sqrt {c+d x}}{b^4}+\frac {(2 b c-7 a d) (c+d x)^{3/2}}{3 b^3}+\frac {(2 b c-7 a d) (c+d x)^{5/2}}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}-\frac {(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{9/2}} \]

[Out]

1/3*(-7*a*d+2*b*c)*(d*x+c)^(3/2)/b^3+1/5*(-7*a*d+2*b*c)*(d*x+c)^(5/2)/b^2/(-a*d+b*c)+a*(d*x+c)^(7/2)/b/(-a*d+b
*c)/(b*x+a)-(-7*a*d+2*b*c)*(-a*d+b*c)^(3/2)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(9/2)+(-7*a*d+2*
b*c)*(-a*d+b*c)*(d*x+c)^(1/2)/b^4

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Rubi [A]
time = 0.08, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 52, 65, 214} \begin {gather*} -\frac {(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{9/2}}+\frac {\sqrt {c+d x} (2 b c-7 a d) (b c-a d)}{b^4}+\frac {(c+d x)^{3/2} (2 b c-7 a d)}{3 b^3}+\frac {(c+d x)^{5/2} (2 b c-7 a d)}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (a+b x) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^(5/2))/(a + b*x)^2,x]

[Out]

((2*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^4 + ((2*b*c - 7*a*d)*(c + d*x)^(3/2))/(3*b^3) + ((2*b*c - 7*a*d)
*(c + d*x)^(5/2))/(5*b^2*(b*c - a*d)) + (a*(c + d*x)^(7/2))/(b*(b*c - a*d)*(a + b*x)) - ((2*b*c - 7*a*d)*(b*c
- a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(9/2)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {x (c+d x)^{5/2}}{(a+b x)^2} \, dx &=\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}+\frac {(2 b c-7 a d) \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{2 b (b c-a d)}\\ &=\frac {(2 b c-7 a d) (c+d x)^{5/2}}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}+\frac {(2 b c-7 a d) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac {(2 b c-7 a d) (c+d x)^{3/2}}{3 b^3}+\frac {(2 b c-7 a d) (c+d x)^{5/2}}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}+\frac {((2 b c-7 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b^3}\\ &=\frac {(2 b c-7 a d) (b c-a d) \sqrt {c+d x}}{b^4}+\frac {(2 b c-7 a d) (c+d x)^{3/2}}{3 b^3}+\frac {(2 b c-7 a d) (c+d x)^{5/2}}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}+\frac {\left ((2 b c-7 a d) (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^4}\\ &=\frac {(2 b c-7 a d) (b c-a d) \sqrt {c+d x}}{b^4}+\frac {(2 b c-7 a d) (c+d x)^{3/2}}{3 b^3}+\frac {(2 b c-7 a d) (c+d x)^{5/2}}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}+\frac {\left ((2 b c-7 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^4 d}\\ &=\frac {(2 b c-7 a d) (b c-a d) \sqrt {c+d x}}{b^4}+\frac {(2 b c-7 a d) (c+d x)^{3/2}}{3 b^3}+\frac {(2 b c-7 a d) (c+d x)^{5/2}}{5 b^2 (b c-a d)}+\frac {a (c+d x)^{7/2}}{b (b c-a d) (a+b x)}-\frac {(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 167, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c+d x} \left (105 a^3 d^2+10 a^2 b d (-17 c+7 d x)+a b^2 \left (61 c^2-118 c d x-14 d^2 x^2\right )+2 b^3 x \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )}{15 b^4 (a+b x)}-\frac {\sqrt {-b c+a d} \left (2 b^2 c^2-9 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(c + d*x)^(5/2))/(a + b*x)^2,x]

[Out]

(Sqrt[c + d*x]*(105*a^3*d^2 + 10*a^2*b*d*(-17*c + 7*d*x) + a*b^2*(61*c^2 - 118*c*d*x - 14*d^2*x^2) + 2*b^3*x*(
23*c^2 + 11*c*d*x + 3*d^2*x^2)))/(15*b^4*(a + b*x)) - (Sqrt[-(b*c) + a*d]*(2*b^2*c^2 - 9*a*b*c*d + 7*a^2*d^2)*
ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(9/2)

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Maple [A]
time = 0.13, size = 219, normalized size = 1.23

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a b d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+6 a^{2} d^{2} \sqrt {d x +c}-8 a b c d \sqrt {d x +c}+2 b^{2} c^{2} \sqrt {d x +c}}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{b \left (d x +c \right )+a d -b c}+\frac {\left (7 a^{3} d^{3}-16 a^{2} b c \,d^{2}+11 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{4}}\)	\(219\)
default	\(\frac {\frac {2 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a b d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+6 a^{2} d^{2} \sqrt {d x +c}-8 a b c d \sqrt {d x +c}+2 b^{2} c^{2} \sqrt {d x +c}}{b^{4}}-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{b \left (d x +c \right )+a d -b c}+\frac {\left (7 a^{3} d^{3}-16 a^{2} b c \,d^{2}+11 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{4}}\)	\(219\)
risch	\(\frac {2 \left (3 d^{2} b^{2} x^{2}-10 a b \,d^{2} x +11 b^{2} c d x +45 a^{2} d^{2}-70 a b c d +23 b^{2} c^{2}\right ) \sqrt {d x +c}}{15 b^{4}}+\frac {\sqrt {d x +c}\, a^{3} d^{3}}{b^{4} \left (b d x +a d \right )}-\frac {2 \sqrt {d x +c}\, a^{2} d^{2} c}{b^{3} \left (b d x +a d \right )}+\frac {\sqrt {d x +c}\, a d \,c^{2}}{b^{2} \left (b d x +a d \right )}-\frac {7 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a^{3} d^{3}}{b^{4} \sqrt {\left (a d -b c \right ) b}}+\frac {16 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a^{2} d^{2} c}{b^{3} \sqrt {\left (a d -b c \right ) b}}-\frac {11 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a d \,c^{2}}{b^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) c^{3}}{b \sqrt {\left (a d -b c \right ) b}}\)	\(323\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^(5/2)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/b^4*(1/5*(d*x+c)^(5/2)*b^2-2/3*a*b*d*(d*x+c)^(3/2)+1/3*b^2*c*(d*x+c)^(3/2)+3*a^2*d^2*(d*x+c)^(1/2)-4*a*b*c*d
*(d*x+c)^(1/2)+b^2*c^2*(d*x+c)^(1/2))-2/b^4*((-1/2*a^3*d^3+a^2*b*c*d^2-1/2*a*b^2*c^2*d)*(d*x+c)^(1/2)/(b*(d*x+
c)+a*d-b*c)+1/2*(7*a^3*d^3-16*a^2*b*c*d^2+11*a*b^2*c^2*d-2*b^3*c^3)/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)
/((a*d-b*c)*b)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)^(5/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [A]
time = 1.42, size = 450, normalized size = 2.53 \begin {gather*} \left [\frac {15 \, {\left (2 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 7 \, a^{3} d^{2} + {\left (2 \, b^{3} c^{2} - 9 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (6 \, b^{3} d^{2} x^{3} + 61 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \, {\left (11 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} + 2 \, {\left (23 \, b^{3} c^{2} - 59 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt {d x + c}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {15 \, {\left (2 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 7 \, a^{3} d^{2} + {\left (2 \, b^{3} c^{2} - 9 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (6 \, b^{3} d^{2} x^{3} + 61 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \, {\left (11 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} + 2 \, {\left (23 \, b^{3} c^{2} - 59 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)^(5/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

[1/30*(15*(2*a*b^2*c^2 - 9*a^2*b*c*d + 7*a^3*d^2 + (2*b^3*c^2 - 9*a*b^2*c*d + 7*a^2*b*d^2)*x)*sqrt((b*c - a*d)
/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(6*b^3*d^2*x^3 + 61*a*b^2
*c^2 - 170*a^2*b*c*d + 105*a^3*d^2 + 2*(11*b^3*c*d - 7*a*b^2*d^2)*x^2 + 2*(23*b^3*c^2 - 59*a*b^2*c*d + 35*a^2*
b*d^2)*x)*sqrt(d*x + c))/(b^5*x + a*b^4), -1/15*(15*(2*a*b^2*c^2 - 9*a^2*b*c*d + 7*a^3*d^2 + (2*b^3*c^2 - 9*a*
b^2*c*d + 7*a^2*b*d^2)*x)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) - (6*
b^3*d^2*x^3 + 61*a*b^2*c^2 - 170*a^2*b*c*d + 105*a^3*d^2 + 2*(11*b^3*c*d - 7*a*b^2*d^2)*x^2 + 2*(23*b^3*c^2 -
59*a*b^2*c*d + 35*a^2*b*d^2)*x)*sqrt(d*x + c))/(b^5*x + a*b^4)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1433 vs. \(2 (160) = 320\).
time = 180.12, size = 1433, normalized size = 8.05 \begin {gather*} \frac {2 a^{4} d^{4} \sqrt {c + d x}}{2 a^{2} b^{4} d^{2} - 2 a b^{5} c d + 2 a b^{5} d^{2} x - 2 b^{6} c d x} - \frac {a^{4} d^{4} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{4}} + \frac {a^{4} d^{4} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{4}} - \frac {6 a^{3} c d^{3} \sqrt {c + d x}}{2 a^{2} b^{3} d^{2} - 2 a b^{4} c d + 2 a b^{4} d^{2} x - 2 b^{5} c d x} + \frac {3 a^{3} c d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{3}} - \frac {3 a^{3} c d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{3}} - \frac {8 a^{3} d^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{5} \sqrt {\frac {a d}{b} - c}} + \frac {6 a^{2} c^{2} d^{2} \sqrt {c + d x}}{2 a^{2} b^{2} d^{2} - 2 a b^{3} c d + 2 a b^{3} d^{2} x - 2 b^{4} c d x} - \frac {3 a^{2} c^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{2}} + \frac {3 a^{2} c^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b^{2}} + \frac {18 a^{2} c d^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{4} \sqrt {\frac {a d}{b} - c}} + \frac {6 a^{2} d^{2} \sqrt {c + d x}}{b^{4}} - \frac {2 a c^{3} d \sqrt {c + d x}}{2 a^{2} b d^{2} - 2 a b^{2} c d + 2 a b^{2} d^{2} x - 2 b^{3} c d x} + \frac {a c^{3} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b} - \frac {a c^{3} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 b} - \frac {12 a c^{2} d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{3} \sqrt {\frac {a d}{b} - c}} - \frac {8 a c d \sqrt {c + d x}}{b^{3}} - \frac {4 a d \left (c + d x\right )^{\frac {3}{2}}}{3 b^{3}} + \frac {2 c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 c^{2} \sqrt {c + d x}}{b^{2}} + \frac {2 c \left (c + d x\right )^{\frac {3}{2}}}{3 b^{2}} + \frac {2 \left (c + d x\right )^{\frac {5}{2}}}{5 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)**(5/2)/(b*x+a)**2,x)

[Out]

2*a**4*d**4*sqrt(c + d*x)/(2*a**2*b**4*d**2 - 2*a*b**5*c*d + 2*a*b**5*d**2*x - 2*b**6*c*d*x) - a**4*d**4*sqrt(
-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b*
*2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**4) + a**4*d**4*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2
*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**
3)) + sqrt(c + d*x))/(2*b**4) - 6*a**3*c*d**3*sqrt(c + d*x)/(2*a**2*b**3*d**2 - 2*a*b**4*c*d + 2*a*b**4*d**2*x
- 2*b**5*c*d*x) + 3*a**3*c*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*
b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**3) - 3*a**3*c
*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)
**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**3) - 8*a**3*d**3*atan(sqrt(c + d*x)/sqrt(
a*d/b - c))/(b**5*sqrt(a*d/b - c)) + 6*a**2*c**2*d**2*sqrt(c + d*x)/(2*a**2*b**2*d**2 - 2*a*b**3*c*d + 2*a*b**
3*d**2*x - 2*b**4*c*d*x) - 3*a**2*c**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)*
*3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**2)
+ 3*a**2*c**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/
(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**2) + 18*a**2*c*d**2*atan(sq
rt(c + d*x)/sqrt(a*d/b - c))/(b**4*sqrt(a*d/b - c)) + 6*a**2*d**2*sqrt(c + d*x)/b**4 - 2*a*c**3*d*sqrt(c + d*x
)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3*c*d*x) + a*c**3*d*sqrt(-1/(b*(a*d - b*c)**3))*log(-
a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*
c)**3)) + sqrt(c + d*x))/(2*b) - a*c**3*d*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)
) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b) - 12*
a*c**2*d*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b**3*sqrt(a*d/b - c)) - 8*a*c*d*sqrt(c + d*x)/b**3 - 4*a*d*(c +
d*x)**(3/2)/(3*b**3) + 2*c**3*atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b**2*sqrt(a*d/b - c)) + 2*c**2*sqrt(c + d*x
)/b**2 + 2*c*(c + d*x)**(3/2)/(3*b**2) + 2*(c + d*x)**(5/2)/(5*b**2)

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Giac [A]
time = 0.49, size = 240, normalized size = 1.35 \begin {gather*} \frac {{\left (2 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 16 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{4}} + \frac {\sqrt {d x + c} a b^{2} c^{2} d - 2 \, \sqrt {d x + c} a^{2} b c d^{2} + \sqrt {d x + c} a^{3} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{4}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{8} + 5 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{8} c + 15 \, \sqrt {d x + c} b^{8} c^{2} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{7} d - 60 \, \sqrt {d x + c} a b^{7} c d + 45 \, \sqrt {d x + c} a^{2} b^{6} d^{2}\right )}}{15 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)^(5/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

(2*b^3*c^3 - 11*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 7*a^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-
b^2*c + a*b*d)*b^4) + (sqrt(d*x + c)*a*b^2*c^2*d - 2*sqrt(d*x + c)*a^2*b*c*d^2 + sqrt(d*x + c)*a^3*d^3)/(((d*x
+ c)*b - b*c + a*d)*b^4) + 2/15*(3*(d*x + c)^(5/2)*b^8 + 5*(d*x + c)^(3/2)*b^8*c + 15*sqrt(d*x + c)*b^8*c^2 -
10*(d*x + c)^(3/2)*a*b^7*d - 60*sqrt(d*x + c)*a*b^7*c*d + 45*sqrt(d*x + c)*a^2*b^6*d^2)/b^10

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Mupad [B]
time = 0.45, size = 264, normalized size = 1.48 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{5/2}}{5\,b^2}-\left (\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4}+\frac {\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (\frac {2\,c}{b^2}-\frac {2\,\left (2\,b^2\,c-2\,a\,b\,d\right )}{b^4}\right )}{b^2}\right )\,\sqrt {c+d\,x}-\left (\frac {2\,c}{3\,b^2}-\frac {2\,\left (2\,b^2\,c-2\,a\,b\,d\right )}{3\,b^4}\right )\,{\left (c+d\,x\right )}^{3/2}+\frac {\sqrt {c+d\,x}\,\left (a^3\,d^3-2\,a^2\,b\,c\,d^2+a\,b^2\,c^2\,d\right )}{b^5\,\left (c+d\,x\right )-b^5\,c+a\,b^4\,d}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (7\,a\,d-2\,b\,c\right )\,\sqrt {c+d\,x}}{7\,a^3\,d^3-16\,a^2\,b\,c\,d^2+11\,a\,b^2\,c^2\,d-2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (7\,a\,d-2\,b\,c\right )}{b^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + d*x)^(5/2))/(a + b*x)^2,x)

[Out]

(2*(c + d*x)^(5/2))/(5*b^2) - ((2*(a*d - b*c)^2)/b^4 + ((2*b^2*c - 2*a*b*d)*((2*c)/b^2 - (2*(2*b^2*c - 2*a*b*d
))/b^4))/b^2)*(c + d*x)^(1/2) - ((2*c)/(3*b^2) - (2*(2*b^2*c - 2*a*b*d))/(3*b^4))*(c + d*x)^(3/2) + ((c + d*x)
^(1/2)*(a^3*d^3 + a*b^2*c^2*d - 2*a^2*b*c*d^2))/(b^5*(c + d*x) - b^5*c + a*b^4*d) - (atan((b^(1/2)*(a*d - b*c)
^(3/2)*(7*a*d - 2*b*c)*(c + d*x)^(1/2))/(7*a^3*d^3 - 2*b^3*c^3 + 11*a*b^2*c^2*d - 16*a^2*b*c*d^2))*(a*d - b*c)
^(3/2)*(7*a*d - 2*b*c))/b^(9/2)

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